# A Basic Result on the Theory of Subresultants

Akritas, Alkiviadis G.; Malaschonok, Gennadi I.; Vigklas, Panagiotis S.

Serdica Journal of Computing (2016)

- Volume: 10, Issue: 1, page 031-048
- ISSN: 1312-6555

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topAkritas, Alkiviadis G., Malaschonok, Gennadi I., and Vigklas, Panagiotis S.. "A Basic Result on the Theory of Subresultants." Serdica Journal of Computing 10.1 (2016): 031-048. <http://eudml.org/doc/289533>.

@article{Akritas2016,

abstract = {Given the polynomials f, g ∈ Z[x] the main result of our paper,
Theorem 1, establishes a direct one-to-one correspondence between the
modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g
computed in Q[x], on one hand, and the subresultant prs of f, g computed
by determinant evaluations in Z[x], on the other.
An important consequence of our theorem is that the signs of Euclidean
and modified Euclidean prs’s - computed either in Q[x] or in Z[x] -
are uniquely determined by the corresponding signs of the subresultant prs’s.
In this respect, all prs’s are uniquely "signed".
Our result fills a gap in the theory of subresultant prs’s. In order to place
Theorem 1 into its correct historical perspective we present a brief historical
review of the subject and hint at certain aspects that need - according to
our opinion - to be revised.
ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.},

author = {Akritas, Alkiviadis G., Malaschonok, Gennadi I., Vigklas, Panagiotis S.},

journal = {Serdica Journal of Computing},

keywords = {Euclidean Algorithm; Euclidean Polynomial Remainder Sequence (prs); Modified Euclidean prs; Subresultant prs; Modified Subresultant prs; Sylvester Matrices; Bezout Matrix; Sturm’s prs},

language = {eng},

number = {1},

pages = {031-048},

publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},

title = {A Basic Result on the Theory of Subresultants},

url = {http://eudml.org/doc/289533},

volume = {10},

year = {2016},

}

TY - JOUR

AU - Akritas, Alkiviadis G.

AU - Malaschonok, Gennadi I.

AU - Vigklas, Panagiotis S.

TI - A Basic Result on the Theory of Subresultants

JO - Serdica Journal of Computing

PY - 2016

PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences

VL - 10

IS - 1

SP - 031

EP - 048

AB - Given the polynomials f, g ∈ Z[x] the main result of our paper,
Theorem 1, establishes a direct one-to-one correspondence between the
modified Euclidean and Euclidean polynomial remainder sequences (prs’s) of f, g
computed in Q[x], on one hand, and the subresultant prs of f, g computed
by determinant evaluations in Z[x], on the other.
An important consequence of our theorem is that the signs of Euclidean
and modified Euclidean prs’s - computed either in Q[x] or in Z[x] -
are uniquely determined by the corresponding signs of the subresultant prs’s.
In this respect, all prs’s are uniquely "signed".
Our result fills a gap in the theory of subresultant prs’s. In order to place
Theorem 1 into its correct historical perspective we present a brief historical
review of the subject and hint at certain aspects that need - according to
our opinion - to be revised.
ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2.

LA - eng

KW - Euclidean Algorithm; Euclidean Polynomial Remainder Sequence (prs); Modified Euclidean prs; Subresultant prs; Modified Subresultant prs; Sylvester Matrices; Bezout Matrix; Sturm’s prs

UR - http://eudml.org/doc/289533

ER -

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